A pyramid has a square base. The side of a square is 12 cm and height of the pyramid is 21 cm. The pyramid is cut into 3 parts by 2 cuts parallel to its base. The cuts are at heigh... A pyramid has a square base. The side of a square is 12 cm and height of the pyramid is 21 cm. The pyramid is cut into 3 parts by 2 cuts parallel to its base. The cuts are at height of 7 cm and 14 cm respectively from the base. What is the difference (in cm³) in the volume of top most and bottom most part? Read more

Steps to Solve

1. Calculate the total volume of the pyramid

The volume $V$ of a pyramid is given by the formula:

$$ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} $$

For the given pyramid, the base is a square with a side length of 12 cm. Thus, the base area $A$ is:

$$ A = 12 \times 12 = 144 \text{ cm}^2 $$

Substituting into the volume formula:

$$ V = \frac{1}{3} \times 144 \times 21 $$

2. Calculate the volume of the topmost part

To find the volume of the topmost section of the pyramid (from height 14 cm to the top of the pyramid), we need the height of this smaller pyramid:

Height of the smaller pyramid = Total height - Height of cut = $21 - 14 = 7$ cm.

The ratio of the side lengths of similar pyramids is the same as the ratio of their heights, thus:

$$ \text{Side length}_{top} = \frac{7}{21} \times 12 = 4 \text{ cm} $$

Now we calculate the volume of the smaller pyramid (topmost part):

$$ A_{top} = 4 \times 4 = 16 \text{ cm}^2 $$

So, the volume of the topmost part is:

$$ V_{top} = \frac{1}{3} \times 16 \times 7 $$

3. Calculate the volume of the bottommost part

Now we need the volume of the bottommost section (from height 0 to 7 cm). The height of this section is 7 cm.

The side length for the bottommost pyramid remains the same (12 cm):

$$ A_{bottom} = 12 \times 12 = 144 \text{ cm}^2 $$

Thus, its volume is:

$$ V_{bottom} = \frac{1}{3} \times 144 \times 7 $$

4. Determine the difference in volumes

Finally, we find the difference in volume between the bottommost and topmost parts:

$$ \text{Difference} = V_{bottom} - V_{top} $$

Now we will perform the calculations based on the equations derived.